Search   Memberlist   Usergroups
 Page 4 of 4 [53 Posts] View previous topic :: View next topic Goto page:  Previous  1, 2, 3, 4
Author Message
Tim Golden
science forum Guru Wannabe

Joined: 12 May 2005
Posts: 176

Posted: Wed Jul 19, 2006 12:53 pm    Post subject: Re: Rotations - why are they not vectors

 Quote: "Timothy Golden BandTechnology.com" wrote in message news:1153276468.410161.56060@75g2000cwc.googlegroups.com... Rotation seems to be a product relationship. The complex numbers perform geometrical rotation via product. Your winding concept is pretty easily expressed in the polar coordinate system but what about for a value a+bi? As I read around about winding number http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html I see that it is mostly used in the context of paths. But should it apply to the complex plane? I would reccommend that you discard conventional notions such as complex numbers & polar co-ordinates as mathematical impositions if you wish to continue making progress.

Well, I am able to derive the complex numbers as P3 in the polysign
domain so I do see them as natural and as an integral part of
spacetime. The construction is starkly different from the usual
development. In effect a few polysign definitions( the identity law and
the product) generate the real numbers for n = 2 and complex numbers
for n = 3. There are higher sign systems as well and they have the same
notion of arithmetic product but the law
| A B | = | A || B |
is broken beyond P3. Still all of these systems obey the usual
associative, commutative, and distributive laws that the real and
complex numbers posess.
P1 is congruent with time and so the family can be argued to derive
spacetime as
P1 P2 P3
This argument relies upon the product being fundamentally involved in
physical processes since that is what causes the breakpoint behavior
beyond P3. At a macro scale we see rotation and planar features exist
in the physical environment. That they are exhibited by some
fundamental math as well should not be overlooked.

-Tim
Tim Golden
science forum Guru Wannabe

Joined: 12 May 2005
Posts: 176

Posted: Wed Jul 19, 2006 4:15 pm    Post subject: Re: Rotations - why are they not vectors

Narcoleptic Insomniac wrote:
 Quote: On Jul 18, 2006 9:34 PM CT, Timothy Golden wrote: Terry Padden wrote: Most of what you write is correct - i.e. heading in the right direction - except for that "Pi" stuff; but remember there are innocent mathematicians reading this and we don't want to frighten them. However if you attach the magic phrase "forgetful functors" to what you have written they will mistake you for one of their senior wizards and let you pass without attacking you - unlike what they try on me. For now I am just trying to get them to think about vectors. It is not easy. They are taught so much rubbish it is hard for them to think clearly. You ask them a simple, but not easy, question about vector space axioms and they - with one or two honourable exceptions -automatically talk about groups, matrices, Rn, Euclidean space, Lie Algebras, "pi", and god knows what; any old rubbish except considering and responding to the question ; and of course they think they are really smart mathematicians - god help us ! *bites tounge* Rotation seems to be a product relationship. The complex numbers perform geometrical rotation via product. Indeed they do. Your winding concept is pretty easily expressed in the polar coordinate system but what about for a value a+bi? As I read around about winding number http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html I see that it is mostly used in the context of paths. But should it apply to the complex plane? Here is one way to implement winding counters in them. So for example rather than a+bi lets just look at just the a part. a i i i i i = a i would be the ordinary notation. But to implement counting we need to add a component to the elemental structure like: ( 0, a ) i i i i i = 1, a i or ( 2, 1.2 )( 3, 2.0 i ) = 5, 2.4 i and ( 0 , -i )( 0 , -i ) = 1, i . It seems you are grasping at a way to determine what branch you're working within after a given set of rotations. The question arises wether these are nondecreasing as more products are taken, or can they unwind? I think they are nondecreasing; they are coupled to the handedness of the coordinate system. So the discrete product would produce a discontinuity on the positive real axis, where a slight change toward -i from a slightly positive i value will cause an abrupt change, though the usual math is undisturbed. Again, it seems that you're thinking about moving from one branch to the next. By branch do you mean the lines dividing the quadrants?

But we can take a sum of these values and get off of the branches.
The counters would generally track but for instance
( ( 0, 3 ) + ( 0, -3 i ) )( 0, i )
= ( 0, 3i ) + ( 1, 3 )
So that independent counters are needed even though generally a
variable z could be construed to have a singular counter. The z counter
can be seen in the polar domain as added directly to the angle. What
this implies about negative angles I'm not quite sure. That would need
some thought. I think that this system asks that all continuous motion
be taken in a singular angular direction. It's the only coherent
behavior. In a physics regard this could be acceptable since the

 Quote: From a physical point of view this may be meaningful since handedness plays a part in physics. If an iterative product raises the winding count should an iterative division process bottom out? That would imply disallowing negative winding numbers which might be a coherent choice. It all seems contrived but might be worth keeping an awareness of these options. - Tim Hmmmmm I dunno, one guy who tried to generalize this kind of geometry of complex numbers ended up inventing the quaternions...

Ha,ha. I'm surprised he didn't wind up with polysigned numbers.
They are quite a different beast yet have the same common features of
real and complex numbers in their midst.
But you are suggesting that winding number in particular could have
some interesting effects. The puzzle might be one of hunting for these
things in the physical. They are suggestive of a lattice structure,
where the winding portion is like a discrete lattice and the continuous
part is the continuum between. That's fairly close to the traditional
buildout of the real numbers; integers then continuum. Perhaps we
should just leave it a can of worms up on the shelf rather than open it
up. Letting out all those worms could be a lot of work to clean up.

 Quote: ..and we all know how horrible that turned out to be (their imaginary parts can describe 3-D rotation and the whole concept leaked into groups, matrices, and god knows what).

Another way to look at rotation is in terms of point fixing. So in 2D
upon fixing a point there is one dimension of freedom to rotate in.
Likewise in 3D upon fixing 2 points (forming an axis) there is one
degree of freedom. Fixing one point in 3D leaves two degrees of
freedom. So in n dimensions there are n-m degrees of freedom, where m
is the number of fixed points. I understand that is not a proof but it
is easily convincing.

This is the traditional sense of rotation instead of allowing for a
real line binary rotation (flip) and then generalizing to scaling also.
In that most general case I think is is simply a matter of allowing
scale in and so the degree of freedom is raised by one to n-m+1. When
the scale is negative things get their handedness flipped.

But all of this does go on in vector projection right? so an
n-dimensional system is put though an nxn matrix multiplication and out
comes a new image. If the matrix was orthogonal and unitary (probably
not the right lingo; something about a determinant here) then the
result conserves its shape. Where is the winding in that?

-Tim
Narcoleptic Insomniac
science forum Guru

Joined: 02 May 2005
Posts: 323

Posted: Thu Jul 20, 2006 12:27 am    Post subject: Re: Rotations - why are they not vectors

On Jul 19, 2006 1:15 AM CT, Timothy Golden wrote:

 Quote: Narcoleptic Insomniac wrote: On Jul 18, 2006 9:34 PM CT, Timothy Golden wrote: Terry Padden wrote: Most of what you write is correct - i.e. heading in the right direction - except for that "Pi" stuff; but remember there are innocent mathematicians reading this and we don't want to frighten them. However if you attach the magic phrase "forgetful functors" to what you have written they will mistake you for one of their senior wizards and let you pass without attacking you - unlike what they try on me. For now I am just trying to get them to think about vectors. It is not easy. They are taught so much rubbish it is hard for them to think clearly. You ask them a simple, but not easy, question about vector space axioms and they - with one or two honourable exceptions - automatically talk about groups, matrices, Rn, Euclidean space, Lie Algebras, "pi", and god knows what; any old rubbish except considering and responding to the question ; and of course they think they are really smart mathematicians - god help us ! *bites tounge* Rotation seems to be a product relationship. The complex numbers perform geometrical rotation via product. Indeed they do. Your winding concept is pretty easily expressed in the polar coordinate system but what about for a value a+bi? As I read around about winding number http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html I see that it is mostly used in the context of paths. But should it apply to the complex plane? Here is one way to implement winding counters in them. So for example rather than a+bi lets just look at just the a part. a i i i i i = a i would be the ordinary notation. But to implement counting we need to add a component to the elemental structure like: ( 0, a ) i i i i i = 1, a i or ( 2, 1.2 )( 3, 2.0 i ) = 5, 2.4 i and ( 0 , -i )( 0 , -i ) = 1, i . It seems you are grasping at a way to determine what branch you're working within after a given set of rotations. The question arises wether these are nondecreasing as more products are taken, or can they unwind? I think they are nondecreasing; they are coupled to the handedness of the coordinate system. So the discrete product would produce a discontinuity on the positive real axis, where a slight change toward -i from a slightly positive i value will cause an abrupt change, though the usual math is undisturbed. Again, it seems that you're thinking about moving from one branch to the next. By branch do you mean the lines dividing the quadrants?

No, any complex number, like z = a + bi you were talking
about, can be represented as z = |z| * e^(it) where |z|
is the modulus (a real number) and t is the argument.
The modulus |z| is the distance from the origin and the
argument t is the angle (similar to polar coordinates).

I guess a good way to bring up "branch" here is by
elaborating on that concept you brought up earlier that
multiplication acts as a rotation.

Since i = e^(pi * i / 2) each time you multiply by i you
are rotating pi * i / 2 radians. This implies that

i^5 = e^(5 * pi * i / 2)

...but e^(it) has a period of 2pi so...

5 pi / 2 == pi / 2 (mod 2pi)

...which gives us i^5 = e^(pi * i / 2) = i.

Basically this says a rotation of 90 degress is the same
as a rotation of 360 + 90 (...but we wanted to avoid
this...)

Okay, what I'm getting at is that this periodicity of
e^(it) can be a problem when we deal with multivalued
functions (e.g. the complex log). To get rid of the
problem we define a specific "branch" to work on. This
just means that we specify a period of 2pi to work within.
For instance, the principal branch is -pi < t <= pi, see

http://mathworld.wolfram.com/PrincipalBranch.html

...for a pretty picture. So why the hell did I even bring
this all up? Because when you wrote...

 Quote: ( 0, a ) i i i i i = 1, a i

...above the "1" just signified that we moved up one
branch.

 Quote: But we can take a sum of these values and get off of the branches. The counters would generally track but for instance ( ( 0, 3 ) + ( 0, -3 i ) )( 0, i ) = ( 0, 3i ) + ( 1, 3 ) So that independent counters are needed even though generally a variable z could be construed to have a singular counter. The z counter can be seen in the polar domain as added directly to the angle. What this implies about negative angles I'm not quite sure. That would need some thought.

Just as positive arguments made us jump up branches,
negative arguments would make us jump down branches.

 Quote: I think that this system asks that all continuous motion be taken in a singular angular direction. It's the only coherent behavior. In a physics regard this could be acceptable since the complex plane already posesses perfect symmetry about the real axis. From a physical point of view this may be meaningful since handedness plays a part in physics. If an iterative product raises the winding count should an iterative division process bottom out? That would imply disallowing negative winding numbers which might be a coherent choice. It all seems contrived but might be worth keeping an awareness of these options. - Tim Hmmmmm I dunno, one guy who tried to generalize this kind of geometry of complex numbers ended up inventing the quaternions... Ha,ha. I'm surprised he didn't wind up with polysigned numbers. They are quite a different beast yet have the same common features of real and complex numbers in their midst. But you are suggesting that winding number in particular could have some interesting effects. The puzzle might be one of hunting for these things in the physical. They are suggestive of a lattice structure, where the winding portion is like a discrete lattice and the continuous part is the continuum between. That's fairly close to the traditional buildout of the real numbers; integers then continuum. Perhaps we should just leave it a can of worms up on the shelf rather than open it up. Letting out all those worms could be a lot of work to clean up.

I have yet to open the can of polysigned worms; looks
pretty interesting though.

 Quote: ..and we all know how horrible that turned out to be (their imaginary parts can describe 3-D rotation and the whole concept leaked into groups, matrices, and god knows what). Another way to look at rotation is in terms of point fixing. So in 2D upon fixing a point there is one dimension of freedom to rotate in. Likewise in 3D upon fixing 2 points (forming an axis) there is one degree of freedom. Fixing one point in 3D leaves two degrees of freedom. So in n dimensions there are n-m degrees of freedom, where m is the number of fixed points. I understand that is not a proof but it is easily convincing. This is the traditional sense of rotation instead of allowing for a real line binary rotation (flip) and then generalizing to scaling also. In that most general case I think is is simply a matter of allowing scale in and so the degree of freedom is raised by one to n-m+1. When the scale is negative things get their handedness flipped. But all of this does go on in vector projection right? so an n-dimensional system is put though an nxn matrix multiplication and out comes a new image. If the matrix was orthogonal and unitary (probably not the right lingo; something about a determinant here) then the result conserves its shape. Where is the winding in that? -Tim

Hooooold on a minute, now you're talking about using
matrices...

matrices (or virtually anything else) can describe
rotations. I immediately brought that up (along with a
few others) and was deemed a "fool" and told to get lost
until I could understand the axioms. I think I shall take
that advice and get out of here.

Regards,
Kyle Czarnecki
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Thu Jul 20, 2006 10:33 am    Post subject: Re: Rotations - why are they not vectors

"ben" <benedict.williams@gmail.com> wrote in message
 Quote: I'm quite happy with my understanding of rotations, which is that they are orientation preserving linear isometries of a vector space. If you want to use a different definition of rotation from me, go ahead. I think it is very likely that rotations (in your sense of the word) form a vector space. I would be greatly pleased if you could offer a rigorous definition of what you understand by "a rotation", as this might allow me to say for certain. My guess is that a rotation, for you, is a measurement along with the implicit understanding of what this rotation might do if you applied it to something,

Of course you are happy with the mathematical status quo. "happy as a pig
in swill" is the relevant proverb. It means you can be mentally lazy and
not think for yourself. Have you not heard of and digested "the Parable of
the talents" ?

However you have been showing some signs of being able to think for yourself
so I will try to be patient with you - it's not easy but we all have to make
sacrifices - even me !

By assuming your definition you are unable to even consider my question. You
have already defined rotations as operators! Now I am aware that the
mathematical high church dictates that rotations are operators - hence all
that matrix jazz you people blindly chant whenever an unbeliever is sighted
; but a heretic like me does not have to worry about your sacred gospel.

Think of it this way - and try not to jump your fences - take them one at a
time. In vector space theory there are (a) vectors = passive things; and
(b) operators = active things which can operate on the (a)'s. Hereabouts
you may need to use the forgetful functor whatsit so you can disregard
Einstein's credo that it is wrong to assume such an active / passive
distinction.

Then reword my question like so. We think of linear displacements as
passive = vectors (a); why can't we think of angular displacements as also
being passive i.e. as vectors ? (This is where the axioms come in). Why
are you forced to think of them as (b) ?

Remember one question at a time. It may help to free up your thought
patterns if you consider that angular displacements are only linear
displacements for rotations of infinite radius.
Shmuel (Seymour J.) Metz1
science forum Guru

Joined: 03 May 2005
Posts: 604

Posted: Thu Jul 20, 2006 11:00 am    Post subject: Re: Rotations - why are they not vectors

07/16/2006
at 09:31 PM, jw12jw12jw12@yahoo.com said:

 Quote: 1-D rotations ARE a vector space...this space is isomorphic to R^1.

No.

 Quote: For example, one of the axioms of a vector space is: u+v=v+u

It's not enough to satisfy one axiom, you must satisfy them all.

 Quote: and in the case of 1-D rotations a rotation of (for example) 30 degrees followed by a rotation of 40 degrees is equivalent to a rotation of 40 degrees follwed by a rotation of 30 degrees.

And what is a rotation of 180° followed by a rotation of 180°?

 Quote: All other axioms woukd be satisfied in this interpretation.

No. See above.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
William Elliot
science forum Guru

Joined: 24 Mar 2005
Posts: 1906

Posted: Thu Jul 20, 2006 11:30 am    Post subject: [] Rotations - why are they not vectors

On Thu, 20 Jul 2006, Terry Padden wrote:
 Quote: However you have been showing some signs of being able to think for yourself so I will try to be patient with you - it's not easy but we all have to make sacrifices - even me ! Oh pity poor pathetic down talking Terry, Padden is superfical ego

by grunting great grandious sacrifices made just for me and y'all.
mariano.suarezalvarez@gma

Joined: 28 Apr 2006
Posts: 58

Posted: Thu Jul 20, 2006 3:53 pm    Post subject: Re: Rotations - why are they not vectors

 Quote: By assuming your definition you are unable to even consider my question. You have already defined rotations as operators!

no indication that you were using the words "vector"
and "rotation" with a meaning different from what the
mathematical community has used them for the last hundred
years (of course, you are free to do that, and you obviously do not
need me to tell you so). I fail to see on what base you imagined

Moreover, when you realised that most people had not
telepathically picked the fact that you were using definitions
different from the usual ones, you did not provide the definitions
you are using, but called them (us, in fact) idiots and went on
to rant on the mathematical establishment and what not.

I'd appreciate that in the future you were more explicit about
the intended meaning of your words, in the situations---such as
the current one,---in which confusion is quite likely to occur.
Actually, I am quite sure this would be of the most utility to
yourself, as it would, in all likelyhood, reduce the number of
"idiots" that respond to you, so that in the end you'll get a higher
signal-to-noise ratio. For example, if you had clearly stated
that you were using "vector" and "rotation" in the (yet
unspecified) meaning you have in mind, I probably would

Cheers,

-- m
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Fri Jul 21, 2006 10:14 am    Post subject: Re: Rotations - why are they not vectors

"Shmuel (Seymour J.) Metz" <spamtrap@library.lspace.org.invalid> wrote in
message news:44bf7072\$51\$fuzhry+tra\$mr2ice@news.patriot.net...
..
 Quote: And what is a rotation of 180° followed by a rotation of 180°? All other axioms woukd be satisfied in this interpretation. No. See above.

Whatever 180 deg + 180 deg is, it is NOT the Zero Vector for rotations
!!!!!!

By AXIOMATIC definition the Zero Vector is ALWAYs the algebraic sum of
opposing equal vectors

e.g. (Rotate +180 deg) + (Rotate -180deg)

Your total misunderstanding of vectors and of rotations which are
multi-valued entities is showing. Equating any number of complete turns
with no rotation is a disgraceful trick played on innocent young minds by
stupid mathematicians. It is on a par with the rubbish that tells them that
3 x 4 = 4 x 3. Anyone who has ever moved house knows that to be rubbish;
they don't need a Ph D in non-commutative geometry to work it out. Just
proper reasoning based on relevant experience.

If you wish to make some progress start by discarding that 360 degree
rubbish. 1 turn = 1 are the only sensible units..

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 4 of 4 [53 Posts] Goto page:  Previous  1, 2, 3, 4 View previous topic :: View next topic
 The time now is Sat Nov 17, 2018 6:00 pm | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Best fit orthogonal basis for list of vectors chengiz@my-deja.com num-analysis 4 Wed Jul 19, 2006 6:16 pm Null rotations Greg Egan Research 1 Sun Jul 02, 2006 12:08 pm vectors from l_2 and their difference isn't in l_1+uncoun... eugene Math 7 Fri Jun 23, 2006 10:52 pm notation for vectors and points pluton Math 3 Sat Jun 03, 2006 4:15 pm ? e-vectors of sum of rank one matrices Cheng Cosine Math 1 Sat Jun 03, 2006 6:36 am